Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for 8th grade pythagorean theorem problems? Then this Grade 8 lesson on The Pythagorean Theorem and Its Converse explains each idea step by step, with worked examples.
Quick answer: The Pythagorean Theorem says that in a right triangle the squares of the two legs add up to the square of the hypotenuse: a² + b² = c². Its converse also holds, so if three side lengths satisfy this equation, the triangle has a right angle. It finds missing lengths in 2D and 3D.
Standards: 8.G.B.6, 8.G.B.7 · Level: Grade 8
What does the Pythagorean Theorem say?
Name the legs a and b and the hypotenuse c in any right triangle. Then:
a2 + b2 = c2
Side c faces the right angle and is the longest of the three.
Why is the theorem true?
Draw a square on every side of the triangle. The squares on the two legs, put together, have the same area as the one on the hypotenuse. One well-known proof fits four copies of the triangle into a large square in two different layouts. What remains uncovered is a2 + b2 in the first layout and c2 in the second, so the two are equal.
How do you use the converse?
Take any triangle and let c be its longest side. When a2 + b2 = c2 is true, the angle opposite c is a right angle. When it is false, the triangle has no right angle.
Test 20, 21, 29: 20² + 21² = 400 + 441 = 841, and 29² = 841. The triangle is a right triangle.
How do you find a missing side?
- Pick out the hypotenuse, the side across from the right angle, and call it c.
- Substitute the known lengths into a2 + b2 = c2.
- Isolate the unknown squared length and take its positive square root.
- Round when the root is not a whole number, and write the units.
A 25-foot ladder has its base 7 feet from a wall. How high does it reach? The ladder is the hypotenuse, so 72 + h2 = 252. Then h2 = 625 − 49 = 576 and h = 24 feet.
How does it work in three dimensions?
Apply the theorem twice. For a box with length l, width w, and height h, the diagonal of the base is √(l2 + w2), and the space diagonal d satisfies d2 = l2 + w2 + h2. For 12, 9, and 8: 144 + 81 + 64 = 289, so d = 17.
Taking the root of each term separately gives a wrong answer, because √(a2 + b2) is not a + b. Add the squares, then take a single root.
Ask first: is the unknown the longest side? If so, add the squares. If it is a leg, subtract.
- In a right triangle, a2 + b2 = c2, where c is the hypotenuse.
- The hypotenuse is opposite the right angle and is the longest side.
- If a2 + b2 = c2 holds for a triangle, the triangle is right (converse).
- To find a leg, subtract: a2 = c2 − b2.
- The space diagonal of a box satisfies d2 = l2 + w2 + h2.
The Pythagorean Theorem and Its Converse: frequently asked questions
Which side is the hypotenuse?
The hypotenuse is the side opposite the right angle, and it is always the longest side of a right triangle. In the equation it is the one that stands alone on the right side.
Can I use the theorem on any triangle?
No. It holds only for right triangles. For other triangles the sum of the squares of two sides does not match the square of the third side, which is exactly how the converse detects them.
What is a Pythagorean triple?
A Pythagorean triple is a set of three whole numbers that satisfy a2 + b2 = c2, such as 20, 21, 29. Multiplying every number in a triple by the same whole number gives another triple.
How do I find a diagonal inside a box?
First find the diagonal of the base with the theorem, then use that diagonal and the height as the legs of a second right triangle. Equivalently, d2 = l2 + w2 + h2.
Why do I get a decimal for the answer?
Most sums of squares are not perfect squares, so the square root is irrational. Leave the answer as a root for an exact value, or round to the nearest tenth, as in √85 ≈ 9.2.
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